In this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $d\leq 3$. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $d\leq 3$, which preserves the physical properties of the continuos solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system.

翻译：本文研究具有奇异单阱势和退化迁移率的非局部Cahn-Hilliard方程。该方程源于一个更一般模型的特殊情形，该模型描述由肿瘤相和健康相组成的二元、饱和、封闭且不可压缩的混合物在有界域中的演化。该一般系统将平均速度场的Darcy型演化与营养物浓度的对流-反应-扩散型演化以及肿瘤相的非局部对流Cahn-Hilliard方程相耦合。主要的数学难点在于证明Cahn-Hilliard方程中肿瘤相的分离性质：据我们所知，该问题在文献中仍属开放性问题。为此，本文仅针对Cahn-Hilliard方程本身进行解析研究。对于具有奇异单阱势和退化迁移率的非局部Cahn-Hilliard方程，我们研究空间维度$d\leq 3$下弱解的存在性和唯一性。在证明存在性后，我们进一步证明三维空间的严格分离性质，该性质同时也适用于低维空间，从而为解的唯性证明铺平道路。最后，针对$d\leq 3$的情形，我们提出一个适定且梯度稳定的连续有限元近似，该近似方法不仅保持了连续解的物理特性且计算高效，并通过二维空间的数值模拟结果验证了所提方案的一致性，同时描述了与系统相关的相序动力学行为。